A circle has a radius of $9$. An arc in this circle has a central angle of $105^\circ$. What is the length of the arc? ${18\pi}$ ${105^\circ}$ $\color{#DF0030}{\dfrac{21}{4}\pi}$ ${9}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (9) = 18\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{105^\circ}{360^\circ} = \dfrac{s}{18\pi}$ $\dfrac{7}{24} = \dfrac{s}{18\pi}$ $\dfrac{7}{24} \times 18\pi = s$ $\dfrac{21}{4}\pi = s$